Predicate Logic; Construct Proofs

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Constructing Proofs Construct a proof for each of the following symbolic arguments or theorems, using any of the inference rules for propositional or predicate logic. You may use direct proof, CP, or RAA. 1. 1. (∃x)(Ax&Bx) ∴ (∃x)Ax&(∃x)Bx 2. 1. (x)(~(Mx∨Nx)→Px) 2. ~Ma 3. ~Na ∴ Pa 3. 1. ~(∃x)(Fx&Gx) ∴ (x)(Fx→~Gx) 4. 1. (x)((Px&Qx)→Rx) 2. Pc 3. ~(∃x)(Px&~Qx) ∴ (∃x)Rx 5. 1. (x)(~Fx∨Hx) 2. (y)(~Gy→~Hy) ∴ (z)(Fz→Gz) 6. ∴ (x)(Px→~Qx)↔(y)(Qy→~Py) Chapter 9 ­ Predicate Logic The system of natural deduction developed in Chapter 8 is powerful, but incomplete. 9.1 The Language of Predicate Logic In order to extend our system of natural deduction to cover categorical logic, we must first develop a method of symbolizing categorical propositions. Predicates, Constants, and Variables Predicate letters are capital letters ­ A through Z ­ used to express properties. Capital letters can still be used to stand for propositions as needed, but we are here expanding their use. Individual constants are lowercase letters ­ a through u ­ used to name individuals. Individual variables are lowercase letters ­ v through z ­ used primarily as placeholders. It is very important to keep in mind the distinction between predicate letters and propositional letters. When a capital letter is coupled with an individual constant or variable, it is a predicate letter that is being used to express a property ­ for example, Fa, Gd, and Px. When a capital letter is not coupled with an individual constant or variable, it is being used to stand for a proposition ­ for instance, F, G, and P. The Universal Quantifier We indicate the scheme of abbreviation in parentheses following the proposition to be symbolized. A quantifier is an expression used to indicate how many things have a given property. Example: For all x, x is human. (Hx: x is human) For the sake of brevity, we will use a variable enclosed in parentheses to stand for the phrase "for all x". Thus, we can symbolize this as: (x)Hx The symbol (x) is called a universal quantifier. It may be read variously as "for any x", "for every x", "for each x", and "for any individual x". Note that we could use a variable other than x, since the variables in the scheme of abbreviation are merely placeholders. So, (y)Hy is also a correct symbolization. The universal quantifier can also be use to symbolize a universal affirmative proposition, such as the following: All humans are mortal. (Hx: x is human; Mx: x is mortal) Let us being by rephrasing this universal affirmative. It say, in effect, that if anything is human, then it is mortal. We can therefore rewrite "All humans are mortal" as follows: For all x, if x human, then x is mortal. The advantage of this technical language, which we will call "logicese", is that it is easily translated into symbols. Using the scheme of abbreviation provided, we can translate the proposition into symbols as follows: (x)(Hx→Mx) Note that universal affirmative propositions involve the arrow. Universal affirmatives can be expressed in a variety of ways in English. universal negative Example: No trees are animals (Tx: x is a tree; Ax: x is an animal) This says, in effect, that if anything is a tree, then it is not an animal. In logicese, we get: For any x, if x is a tree, then x is not an animal. Which we can symbolize as: (x)(Tx→~Ax) There are various ways of expressing universal negatives in English. Note that the placement of tildes is very important when translating universal negative propositions. The Existential Quantifier We now introduce a second quantifier, called the existential quantifier. Our symbol for the existential quantifier looks like this: (∃x). This symbol is read, "There exists some x such that" or simply "For some x". Example: Something is mortal. (Mx: x is mortal) (∃x)Mx The existential quantifier allows us to symbolize particular affirmatives, such as the following: Some dogs are collies. (Dx: x is a dog; Cx: x is a collie) (∃x)(Dx&Cx) When symbolizing particular affirmatives, we need to combine the existential quantifier with the ampersand rather than the arrow. particular negative propositions: Example: Some dogs are not collies. (Dx: x is a dog; Cx: x is a collie) (∃x)(Dx&~Cx) There are numerous ways of expressing particular negatives in English. One final note of caution: The existential quantifier can sometimes be used to translate the word "any", since "any" can sometimes mean "even one". This is clearest in the case of "if any" conditionals like: If anyone complains, Dad is going to turn this car around. logicese: If, for some x (x is a person and x complains), then Dad turns the car around. (Px: x is a person; Cx: x complains; d: Dad; Tx: x turns the car around) (∃x)(Px&Cx)→Td The Language of Predicate Logic: A More Precise Formulation We must have a clear grasp of what counts as a well­formed formula of predicate logic. The vocabulary of predicate logic consists of: propositional letters (capital letters A through Z) individual constants (lowercase letters through u) individual variables (v,w,x,y,z) predicate letters (capital letters A through Z when coupled with individual constants or variables the logical operators the quantifier symbols parentheses An expression of predicate logic is any sequence of symbols in this vocabulary. An atomic formula of predicate logic is either a propositional letter or a predicate letter coupled with individual constants or variables. We will use the capital, cursive letters P and Q to stand for any expressions in the language of predicate logic. And we will use the bold letter x to stand for any individual variable (v, w, x, y, and z). Using this notation we can say that a symbolic expression is a well­formed formula (WFF) of predicate logic under the following conditions: 1. Every atomic formula is a WFF. 2. If P is a WFF, then so is (x)P. 3. If P is a WFF, then so is (∃x)P. 4. If P is a WFF, then so is ~P. 5. If P and Q are WFFs, then so is (P&Q). 6. If P and Q are WFFs, then so is (P∨Q). 7. If P and Q are WFFs, then so is (P→Q). 8. If P and Q are WFFs, then so is (P↔Q). Nothing counts as a WFF of predicate logic unless it can be demonstrated to be one by application of the above conditions. We continue to allow some informal uses for the sake of convenience; for example, we permit the omission of parentheses when no ambiguity results, and brackets may be employed to increase readability. The scope of a quantifier within a formula is the shortest WFF immediately to the right of the quantifier. An occurrence of a variable x is bound if it lies within the scope of an x­quantifier. an "x­quantifier" is a quantifier that has "x" in it, i.e., either (x) or (∃x) An occurrence of a variable is free if it is not bound. Why does it matter whether a variable is bound or free? A propositional function is a wff of predicate logic that includes a free occurrence of a variable. Quantifiers can bind only occurrences of the variables they contain. 9.2 Demonstrating Invalidity An algorithm is a precisely described and finite procedure for solving a problem. The truth tables we studied in Chapter 7 are an algorithm for propositional logic. If we follow the correct procedures for constructing a truth table, we can determine the validity of any argument within propositional logic. Unfortunately, there is no such algorithm for predicate logic. Nevertheless, there are methods similar to truth tables that can be used to evaluate many arguments in predicate logic. We will examine one such method in this section, the finite universe method. An argument is invalid if it is possible for its conclusion to be false while its premises are true. Thus, if we can describe a possible situation in which the conclusion of an argument is false while its premises are true, then we have shown the argument to be invalid. This is the essential principle underlying the finite universe method. And the finite universe method enables us to describe such situations simply and abstractly by imagining universes with a small number of objects. To understand the finite universe method, we must first understand the meaning of quantified propositions in universes containing a small number of objects. Let us first consider the meaning of universally quantified propositions in a two­object universe, containing only two objects ­ a and b. A universally quantified proposition is a WFF of the form (x)P. Because a and b are the only items in this universe, (x)Rx is equivalent in this universe to the following conjunction: Ra&Rb In general, in a finite universe, a universally quantified proposition is equivalent to a certain conjunction. An existentially quantified proposition is a WFF of the form (∃x)P. The proposition, (∃x)Rx is equivalent, in our two­object universe, to the following disjunction: Ra∨Rb In general, in a finite universe, an existentially quantified proposition is equivalent to a certain disjunction. Let us consider a slightly larger universe containing three objects ­ a, b, and c: In this universe, (x)Rx is equivalent to: Ra&(Rb&Rc) In this universe, (∃x)Rx is equivalent to: Ra∨(Rb∨Rc) The general principle should be clear at this point: Universally quantified propositions become conjunctions; existentially quantified propositions become disjunctions. A special case worth noting is that of a universe with only one object, a: In this universe, "Everything is red" is equivalent to "a is red" (in symbols, Ra). But "Something is red" is also equivalent (in our one­object universe) to "a is red) (in symbols, Ra). Thus, in a one­object universe, (x)Rx is equivalent to (∃x)Rx. Two­object universe: universal affirmative: Example: All collies are dogs. (Cx: x is a collie; Dx: x is a dog) In symbols: (x)(Cx→Dx) Equivalent to the conjunction: (Ca→Da)&(Cb→Db) Note the arrow in each conjunct. particular affirmative: Example: Some dogs are collies. (Dx: x is a dog; Cx: x is a collie) In symbols: (∃x)(Dx&Cx) Equivalent to the disjunction: (Da&Ca)∨(Db&Cb) Note the ampersand in each disjunct. To apply the finite universe method: 1. we first translate the premises and conclusion of an argument into atomic formulas (for a one­ object universe) or conjunctions and disjunctions (for a multiple­object universe). 2. We then apply the method of abbreviated truth tables to determine whether the conclusion can be false while the premises are true. The basic idea is that the validity of an argument does not depend on there being a large number of objects in the universe. If a pattern of reasoning allows for true premises and a false conclusion in a two­object universe, then that pattern of reasoning is invalid. Let's try the method out on a short argument: Nothing red is blue. Something is not blue. So, something is red. (Rx: x is red; Bx: x is blue) In symbols: (x)(Rx→~Bx), (∃x)~Bx ∴ (∃x)Rx First translate the premises and conclusion for a one­object universe: Ra→~Ba, ~Ba ∴ Ra Now, we apply the method of abbreviated truth tables: Ra Ba Ra→~Ba, ~Ba ∴ Ra F F F TTF TF F This assignment does the job. We have shown that it is possible for an argument of the preceding pattern to have true premises and a false conclusion. Hence, the form is invalid. A one­object universe will not always be adequate for our purposes. Consider the following argument: Nothing good is evil. Something is good. So, nothing is evil. (Gx: x is good; Ex: x is evil) In symbols: (x)(Gx→~Ex), (∃x)Gx ∴ (x)~Ex For a one­object universe, the argument translates as follows: Ga→~Ea, Ga ∴ ~Ea Now, we apply the abbreviated truth table method: Ga Ea Ga→~Ea, Ga ∴ ~Ea T / FT T FT Although we hypothesized that the premises could be true (while the conclusion is false), we were forced to contradict the hypothesis, as the symbol "/" indicates. So, let's try a two­object universe: Ga Ea Gb Eb (Ga→~Ea)&(Gb→~Eb), Ga∨Gb ∴ ~Ea&~Eb T F F T T T TFT T T TF T T F TFFFT Here, the premises are true and the conclusion is false. This shows that the argument form is invalid. To show that certain kinds of arguments are invalid, we need to consider a universe containing at least three objects. This method becomes rather unwieldy as we consider universes with more than two members. The good news is that in many cases, a one­ or two­object universe will be sufficient to reveal the invalidity of an argument. The bad news here is multiple: 1. there are invalid arguments within predicate logic whose invalidity cannot be shown via the finite universe method. These arguments belong to the logic of relations, the more advanced part of predicate logic (see section 9.5). 2. there are cases in which a large (though finite) universe would be needed to apply the finite universe method, making it impractical without a computer. In spite of these limitations, the finite universe method can deepen our understanding of the meaning of quantified propositions by revealing a great many invalid inferences. In practice, it is usually best to consider a one­object universe first and then try a two­ or three­object universe as needed. Using the finite universe method, we can now sort out some issues that arose in Chapter 5 as we discussed the Aristotelian Square of Opposition: Recall that corresponding categorical propositions have the same subject and predicate terms. And from an Aristotelian perspective, a universal affirmative proposition implies its corresponding particular affirmative ­ for example, "All unicorns are animals" implies "Some unicorns are animals". Similarly, a universal negative proposition implies its corresponding particular negative ­ for example, "No unicorns are horses" implies "Some unicorns are not horses". Modern logicians, following George Boole, deny that these inferences are valid. In general, the inference from a universal affirmative to its corresponding particular affirmative will move from truth to falsehood when the subject terms denote an empty class. The inference from universal negative propositions to their corresponding particular negatives is invalid for the same reason. Here is a related point. In the Aristotelian scheme, corresponding universal affirmative and universal negative propositions are said to be contraries Contraries are propositions that cannot both be true, but can both be false. "All unicorns are animals and "No unicorns are animals", according to Aristotelians, are contraries. But according to modern logicians, these propositions are not contraries because they can both be true. Whenever the subject term of corresponding universal affirmative and universal negative propositions denotes an empty class, both propositions are true. This is why modern logicians deny the Aristotelian thesis that corresponding universal affirmative and universal negative propositions are contraries. A special complication that arises when one quantifier falls within the scope of another: For example: (∃x)(Sx→(y)Ry) ∴ (y)[(∃x)Sx→Ry] How do we translate such formulas for a two­object universe? It may help to translate them in two stages, one quantifier at a time. Let's start with the premise: Stage 1: (Sa→(y)Ry)∨(Sb→(y)Ry) Stage 2: (Sa→[Ra&Rb])∨(Sb→[Ra&Rb]) The translation of the conclusion is as follows: Stage 1: [(∃x)(Sx→Ra]&[(∃x)(Sx→Rb] Stage 2: [(Sa∨Sb)→Ra]&[(Sa∨Sb)→Rb] The fact that the argument is invalid underscores the importance of the scope of the quantifiers, from the standpoint of logic. Summary of the Finite Universe Method 1. 2. 3. 4. 5. Translate the premises and conclusion of the argument into symbols. In a one­object universe, translate all quantified propositions into atomic formulas. In a multiple­object universe, translate universally quantified propositions into conjunctions. In a multiple­object universe, translate existentially quantified propositions into disjunctions. Apply the method of abbreviated truth tables to determine whether it is possible for the conclusion to be false while the premises are true. 9.3 Constructing Proofs All the rules of propositional logic still apply within predicate logic. We will now move beyond propositional logic to add four implicational rules of inference that are specific to predicate logic. To grasp these rules, however, one must first understand what it is to be an instance of a quantified formula. Begin with a quantified formula: (x)[Fx→(∃y)(Gy∨Hx)] Removing the (x) quantifier from the front of this formula leaves us with two free occurrences of the x­variable: Fx→∃y)(Gy∨Hx) This is not a proposition, but a propositional function. However, we can turn it into a proposition by replacing the occurrences of x with an individual constant: Fa→∃y)(Gy∨Ha) We now have an instance of the universally quantified formula we started with. The operation that takes us from a quantified formula to an instance of that formula is called instantiation. The constant used in an instantiation is called the instantial constant. More generally: let the cursive letter P stand for any WFF of predicate logic, the bold letter x stand for any individual variable, and the bold letter c stand for any individual constant. We can then say that an instance of a quantified WFF (x)P or (∃x)P is any WFF obtained by the following steps: Step 1: Remove the initial quantifier, (x) or (∃x) as the case may be. Step 2: In the WFF resulting from Step 1, uniformly replace all free occurrences of the variable x in P with occurrences of c. (We will use Pc to stand for the resulting instance.) There are four important features of this definition: 1. 2. 3. 4. Must remove the initial quantifier. Must uniformly replace all free occurrences of the variable. Must not replace any bound occurrences of variables. Must not replace any occurrences of a variable with another variable. Instantiation always takes us from variables to constants. Universal Instantiation Our first new implicational rule is universal instantiation (UI, for short). We call the rule of inference that permits the move from a universally quantified formula to an instance of that formula "universal instantiation" because it allows us to instantiate a universally quantified formula. There are no restrictions on universal instantiation ­ UI allows us to move from any universally quantified formula to any instance of that formula. We can therefore formulate UI as follows, where P stands for any WFF, x stands for any variable, and c stands for any constant: Universal Instantiation (UI) (x)P ∴ Pc (where Pc is an instance of (x)P) UI is an implicational rule, not an equivalence rule. Hence, UI can be applied only to whole lines in a proof, not to parts thereof. Two final mistakes that are easy to make: 1. Attempting to apply UI to the negation of a universally quantified proposition. 2. Attempting to apply UI to a conditional where the antecedent and consequent are universally quantified propositions. One must exercise extreme caution in classifying formulas because not every formula that includes a quantifier is a quantified formula. Before using UI, you must make sure that you are working with a universally quantified formula. To do this, you must follow two steps: 1. Make sure that a universal quantifier appears at the beginning of the line in question. 2. Make sure that the quantifier has scope over the entire line in question (remember that the scope of a quantifier is the shortest WFF to the right of the quantifier). If a line passes both of these tests, it is a universally quantified formula, in which case you may apply UI. Existential Generalization Our second inferential rule is existential generalization (EG). This rule allows us to existentially generalize an instance of a quantified formula. The operation of generalization is essentially the opposite of instantiation ­ instead of moving from a quantified formula to an instance, we move from an instance to a quantified formula; instead of removing a quantifier and replacing variables with constants, we introduce a quantifier and replace constants with variables. There are no restrictions on the application of EG ­ the rule allows us to infer any existentially quantified formula from any instance of that formula. We can therefore formulate EG as follows, where P stands for any WFF, x stands for any variable, and c stands for any constant: Existential Generalization (EG) Pc ∴ (∃x)P (where Pc is an instance of (∃x)P) As an implicational rule. EG can be applied only to whole lines, so an application of EG should always result in an existentially quantified formula. Let's consider a proof that makes use of both EG and UI. Are both rules applied correctly in this case? 1. (x)Rx ∴(∃x)Rx 2. Rb 1, UI 3. (∃x)Rx 2, EG Yes. Given that everything has the property R, our rules allow us to infer that something is R. This brings out an interesting feature of our system that is shared by other classical systems of logic ­ it includes the assumption that at least one thing exists. Without this assumption, we could not instantiate to b in line (2) because that move assumes that there is at least one individual corresponding to the constant. Existential Instantiation Our third inferential rule is existential instantiation (EI), which allows us to instantiate an existentially quantified formula, subject to certain restrictions. Restrictions: 1. One cannot instantiate to a constant that has already occurred in the proof. 2. One cannot instantiate to a constant that occurs in the conclusion to be proved. We can now formulate EI as follows, where P stands for any WFF, x stands for any variable, and c stands for any constant: Existential Instantiation (EI) (∃x)P ∴ Pc (where Pc is an instance of (∃x)P and c does not occur in an earlier line of the proof or in the last line of the proof.) Tip 1: Apply EI before you apply UI. Universal Generalization Our fourth inferential rule is universal generalization (UG), which allows us to move from an instance of a universally quantified formula to a universally quantified formula, provided that certain conditions are met. Conditions: 1. One cannot generalize from a constant that occurs in a premise of the argument. 2. One cannot generalize from a constant that appears in a line derived by an application of EI. These first two restrictions on UG can be derived from the following prescription: One should only universally generalize from a constant that is introduced by UI. 3. One cannot universally generalize from a constant that appears in the resulting formula. When applying UG, be sure to uniformly replace all occurrences of the relevant constant with free occurrences of a variable before introducing the universal quantifier. We will introduce one further restriction on UG in section 9.4 when we discuss indirect proofs, but for now, we can formulate UG as follows, where P is any WFF, x is any variable, and c is any constant: Universal Generalization (UG) Pc ∴ (x)P (where Pc is an instance of (x)P and c does not occur in a premise of the argument, a previous line derived by an application of EI, or (x)P) 9.4 Quantifier Negation, RAA, and CP In this section, we add an equivalence rule to our system and explain how to use conditional proof and reductio ad absurdum within predicate logic. The rule of quantifier negation (QN) comes in four forms: Quantifier Negation (QN) (∃x)P :: ~(x)~P (∃x)~P :: ~(x)P (x)P :: ~(∃x)~P (x)~P :: ~(∃x)P QN is an equivalence rule, which means that is can be applied to parts of lines in a proof, as well as to whole lines. Tip 2: When a tilde appears on the left side of a quantifier, it is often useful to apply QN and instantiate. Sometimes, the QN rule enables us to make use of propositional logic rules without instantiating. The QN rule is often useful when employing the reductio ad absurdum (RAA) method. RAA and CP can be used in predicate logic, but the use of these methods demands an additional restriction on universal generalization: One cannot universally generalize from a constant that occurs in an undischarged assumption. Thus, our official formulation of UG is as follows, where P stands for any WFF, x stands for any variable, and c stands for any constant: Universal Generalization (UG) Pc ∴ (x)P (where Pc is an instance of (x)P and c does not occur in (a) (x)P), (b) a premise of the argument, (c) a line derived by an application of EI, or (d) an undischarged assumption) Tip 3: If the conclusion is a universally quantified proposition containing an arrow, use CP to prove an instance of the relevant conditional and then apply UG. Tip 4: When the conclusion of an argument is an existentially quantified proposition, RAA is often useful. 9.5 The Logic of Relations: Symbolizations Thus far, we have considered only monadic (one­place) predicate letters, such as Ax, BY, and Cz. Monadic predicate letters are adequate for ascribing an attribute (e.g., being human) to an individual. But, individuals not only have attributes, they also bear relations to one another. For example, we can say that Smith is older than Jones or that Elizabeth is a sister of John. Modern predicate logic encompasses the logic of relations, but to symbolize relations, we need predicate letters with more than one place. These are called polyadic predicate letters. For instance, we can use Oxy to abbreviate "x is older than y" or Sxy to abbreviate "x is a sister of y". Propositions involving relations can be rather difficult to symbolize, but much of the difficulty can be removed by working through a series of well­chosen examples. Perhaps the most important skill to develop here is that of translating English into logicese. Examples English Logicese Symbols Someone loves everyone. (Px: x There is some x such that x is a person, and for all y, if (∃x)[Px & (y) is a person; Lxy: x loves y) y is a person, then x loves y. (Py→Lxy)] Everyone loves someone. For all x, if x is a person, then there is some y such that (x)[Px→(∃y) y is a person and x loves y. (Py&Lxy)] No one loves everyone. For all x, if x is a person, then it is not true that for all y, if y is a person, x loves y. It is not the case that there is some x such that x is a person and for all y, if y is a person, then x loves y. (x)(Px→~(y) (Py→Lxy)] ~(∃x)[Px& (y) (Py→Lxy)] No one loves anyone. For all x, if x is a person, then for all y, if y is a person, (x)[Px→(y) x does not love y. (Py→~Lxy)] Predicate letters can be more than two places. Our symbolizations bring out the logical complexity that can be present in rather ordinary English sentences. The best way to ensure accuracy of translation into symbols within predicate logic is first to translate English into logicese and then to translate logicese into symbols. Certain General Characteristics of Relations A relation R is symmetrical just in case: For all x and y (if x bears R to y, then y bears R to x). Example: being a sibling of A relation R is asymmetrical just in case: For all x and y (if x bears R to y, then it is not the case that y bears R to x). Example: being a mother of A relation R is nonsymmetrical just in case it is neither symmetrical nor asymmetrical. Example: being a sister of A relation R is reflexive just in case: For all x (x bears R to x). Example: identity A relation R is irreflexive just in case: For all x (it is not the case that x bears R to x). Example: being larger than A relation R is nonreflexive just in case it is neither reflexive nor irreflexive. Example: being proud of A relation R is transitive just in case: For all x, y and z (if x bears R to y and y bears R to z, then x bears R to z). Example: being taller than A relation R is intransitive just in case: For all x, y and z (if x bears R to y and y bears R to z, then it is not the case that x bears R to z). Example: being father of A relation R is nontransitive just in case it is neither transitive nor intransitive. Example: being an acquaintance of 9.6 The Logic of Relations: Proofs The inference rules for predicate logic introduced in sections 9.3 and 9.4 are sufficient for the logic of relations. But it will be helpful to highlight some new types of situations that can arise and to issue some important reminders regarding restrictions on our inference rules. First, if you have a premise with more than one quantifier, apply UI or EI to remove the quantifiers one at a time, from left to right. Remember that UI and EI are implicational rules, and hence they cannot be applied to parts of propositions. We can apply UI only to universally quantified propositions. We can apply EI only to existentially quantified propositions. We put quantifiers back on one at a time also. Second, remember that EG and UG are implicational rules. We cannot apply UG to part of a proposition. Third, when using UI, remember that constants must be substituted uniformly. When using EI, constants must be substituted uniformly. Fourth, remember that we may never existentially instantiate to a constant that occurs previously in the proof. Fifth, recall the special restrictions on UG. UG lets us move from Pc to (x)P provided that c does not occur in (i) (x)P (ii) a premise of the argument (iii) a line derived by an application of EI, or (iv) an undischarged assumption The logic of relations enables us to see the subtlety of many rather ordinary­looking English sentences. The order of quantifiers can be very important. 9.7 Identity: Symbolizations Among the many different types of relations, one is particularly important for logic, namely, identity. To introduce identity into our system of logic, we will borrow a symbol from arithmetic, but we will call it the identity sign. Except for the addition of the identity sign, the language for predicate logic with identity is exactly like the language for predicate logic. We will treat identity claims involving individual constants or individual variables as atomic formulas, so there is no need to use parentheses: x=y x=c a=c And we can symbolize negations of identity propositions by means of the negations sign as follows: ~x=y ~x=c ~a=c Note that propositions or propositional functions are formed by placing a constant or a variable on each side of the identity sign. Also note that to negate an identity proposition, we simply attach a tilde to it ­ no parentheses are employed. Thus, to negate a=b, we simply write ~a=b, which is read as "It is not the case that a is identical with b". NOTE: We cannot write: a=~b, as this not a well­formed formula ­ we cannot negate an individual constant or an individual variable. By means of the identity sign we can symbolize many complicated types of propositions. English Logicese Symbols Only Only Edison invented the phonograph. (e: Edison; Px: x invented the phonograph) The only person who is guilty is The Only David. (Px: x is a person; Gx: x is guilty; d: David) No one except Bell invented the telephone. (Px: x is No...Except a person; b: Bell; Tx: x invented the telephone) All European countries except Switzerland declared war. (Ex: x All...Except is a European country; s: Switzerland; Dx: x declared war) e invented the phonograph, and for all x, if x invented Pe&(x)(Px→x=e) the phonograph, then x is identical with e. d is a person and d is guilty, and for all x, if x is a (Pd&Gd)&(x)[(Px&Gx)→x=d] person and x is guilty, then x is identical with d. b is a person and b invented the telephone, and for all x, if x is a person and x (Pb&Tb)&(x)[(Px&Tx)→x=b] invented the telephone, then x is identical with b. s is a European country and s did not declare war, and for all x, if x is a European (Es&~Ds)&(x)[(Ex&~x=s)→Dx] country and x is not identical with s, then x declared war. The tallest mountain is Mount Everest. e is a mountain, and for all (Mx: x is a Superlatives x, if x is a mountain, then e Me&(x)[(Mx&~x=e)→Tex] mountain; Txy: x is is taller than x. taller than y; e: Mount Everest) For all x, for all y, if x is a There is at most one god and y is a god, then x god. (Gx: x is a god) is identical with y. At Most (x)(y)[(Gx&Gy)→x=y] For all x, for all y, for all z, if x is a god and y is a god There are at most and z is a god, then either x (x)(y)(z) two gods. (Gx: x is a is identical with y or x is ([(Gx&Gy)&Gz]→[(x=y∨x=z)∨y=z]) god) identical with z or y is identical with z. There is at least one There exists an x such that utilitarian. (Ux: x is (∃x)Ux x is a utilitarian. a utilitarian) At Least There is some x and there There are at least is some y such that x is a two utilitarians. (Ux: utilitarian and y is a x is a utilitarian) utilitarian and x is distinct from y. There is some x such that x is a solipsist, and for all x, for all y, if x is a solipsist There is exactly one and y is a solipsist, then x Exactly One solipsist. (Sx: x is a is identical with y. solipsist) There is some x such that x is a solipsist, and for all y, if y is a solipsist, then y = x. There is some x and there is some y such that x is a solipsist and y is a solipsist, There are exactly Exactly and x is distinct from y, two solipsists. (Sx: x Two and for all z, if z is a is a solipsist) solipsist, then z is identical with x or z is identical with y. (∃x)(∃y)[(Ux&Uy)&~x=y] (∃x)Sx&(x)(y)[(Sx&Sy)→x=y] (∃x)[Sx&(y)(Sy→y=x)] (∃x)(∃y)([(Sx&Sy)&~x=y]&(z) [Sz→(z=x∨z=y)]) The identity sign has been used to provide an important analysis of definite descriptions. A definite description is an expression of the form "the so­and­so", such as "the smallest prime number", "the discoverer of polonium", or "the author of War and Peace". Such expressions seem intended to denote exactly one object or person. But consider the following example, which Bertrand Russell discussed in "On Denoting": 92. The present King of France is bald. France presently has no king, so the expression "the present King of France" apparently fails to refer to anyone or anything. How then can (92) be a meaningful sentence (as it appears to be)? Russell suggested that statements involving definite descriptions, such as (92) make three claims: a. A thing of a certain type exists (in this case, a present King of France). b. It is unique. c. It has a certain property (in this case, it is bald). We can rewrite (92) in logicese as follows: There is some x such that x is a present King of France, and for all y, if y is a present King of France, then y is identical with x, and x is bald. English The discoverer of Definite polonium is Polish. (Dx: x Descriptions discovered polonium; Px: x is Polish) Logicese There is some x such that x discovered polonium, and for all y, if y discovered polonium, y is identical with x, and x is Polish. Symbols (∃x)(Dx&(y) [(Dy→y=x)&Px]) 9.8 Identity: Proofs To construct proofs involving the identity relation, we will add three new rules of inference. The first of these is called Leibniz' Law. Leibniz' law is the principle that if m an n are identical, then every property of m is a property of n, and vice versa. To state our inference rules in a general fashion, we will use the bold letters m and n to stand for any individual constant. And we will use Pm and Pn to stand for WFF's containing m and n, respectively. Leibniz' law comes in two forms, as follows: Leibniz' Law (LL) m=n n=m Pm Pm ∴Pn ∴Pn Here, we obtain Pn by replacing one or more occurrences of m in Pm with occurrences of n. General principle: if x has a certain property and y lacks that property, then x is not identical to y Our next inference rule is called symmetry (Sm). Where the bold m and n stand for any individual constants, the rule may be stated as follows. It comes in two forms: Symmetry (Sm) m=n ~m=n ∴n=m ∴~n=m Our third and final rule of inference governing the logic of identity is the principle that each thing is identical with itself. We will call this simply the identity (Id) rule. It differs from all our previous rules in that it does not involve a premise. We may represent it as follows: Identity (Id) ∴ n=n Here, the bold n stands for any individual constant. The Id rule allows us to enter propositions of self­identity, such as b=b, as lines in a proof. This rule isn't used very often in constructing proofs, but our system of logic would not be complete without it. Outline
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